Collections have a lists method similar to the method for tables described by @Gadoma. It returns an array containing the value of a given attribute for each item in the collection.
To retrieve the desired array of names from my collection of tags I can simply use:
$tags->lists('name');
Update: As of laravel 5.2 lists is replaced by pluck.
$tags->pluck('name');
More specifically, the laravel 5.2 upgrade guide states that “[t]he lists method on the Collection, query builder and Eloquent query builder objects has been renamed to pluck. The method signature remains the same.”